To solve this problem, we need to analyze the relationship between the vectors u and v, especially since they are at right angles to each other. Given that the horizontal and vertical components of vector u are represented as a and b, we can visualize this in a coordinate system where vector u points from the origin to the point (a, b). Vector v, being perpendicular to u, will have its own components that we need to determine.
Understanding the Vectors
Let's break down the components of the vectors:
- Vector u has a horizontal component of a and a vertical component of b.
- Vector v has a length of d and is perpendicular to vector u.
Using the Pythagorean Theorem
Since vectors u and v are at right angles, we can apply the Pythagorean theorem to find the components of vector v. The length of vector v can be expressed in terms of its horizontal and vertical components. Let's denote the horizontal component of vector v as v_h and the vertical component as v_v.
According to the Pythagorean theorem, we have:
v_h² + v_v² = d²
Finding the Vertical Component
Now, we know that the horizontal component of vector v must be related to the horizontal component of vector u. Since u is at right angles to v, we can express the horizontal component of v as:
v_h = -b (this is because it will be in the opposite direction of the vertical component of u).
Substituting this into our Pythagorean theorem equation gives us:
(-b)² + v_v² = d²
Which simplifies to:
b² + v_v² = d²
Solving for the Vertical Component
Now, we can isolate v_v:
v_v² = d² - b²
Taking the square root of both sides, we find:
v_v = √(d² - b²)
Final Expression
Thus, the vertical component of vector v in terms of a, b, and d is:
v_v = √(d² - b²)
This result shows how the lengths of the components relate to the overall length of vector v, while also considering the right-angle relationship with vector u. It's a great example of how vector components interact in a two-dimensional space!
